The measure of an inscribed angle is half that of its intercepted arc.
Given that the arc ADB measures 270∘, find the measure of the inscribed angles ∠D and ∠C.
To find the angle measures, we can use the Inscribed Angle Theorem. Thus, we have to find the measure of arc AB first. The minor arc AB, together with its corresponding major arc ADB, corresponds to a full rotation. Thus, the sum of their measures is 360∘. This means that the measure of AB must be 360∘−270∘=90∘. By the Inscribed Angles Theorem, angle ∠D measures half that of its intercepted arc, AB.
Thus, m∠D=290∘=45∘. For the same reason, m∠C is 45∘ as well.
When two tangents of a circle intersect to create an angle, it is called a circumscribed angle.
Polol is a newbie chopstick user. When he picks a certain sushi roll up, he notices that he crosses the chopsticks. Out of curiosity, he wants to know the circumscribed angle created by the crossed chopsticks. For this, he makes two measurements: the arc between the chopsticks' point of contact with the roll measures 4.7 cm and the sushi roll's diameter is 3.5 cm. Use his measurements to find the angle.
Let's start by drawing a diagram corresponding to the situation. Notice that the sushi roll can be modeled by a circle, with the chopsticks being tangents to the circle.
The diameter of the circle is then 3.5 cm, the length of arc BC is 4.7 cm, and we are interested in the measure of angle ∠A. To find this, we'll first need the measure of arc BC, which we can find by the ratio between the arc length and the circumference. The circumference is πd=π⋅3.5≈11.0 cm. We can now calculate the arc measure.
We now know that the measure of arc BC is 154∘. As ABOC is a quadrilateral, it has the interior angle sum 360∘. With m∠B and m∠C being 90∘, the sum of m∠A and m∠O must then be 180∘.
The measure of the circumscribed angle is 26∘.
A polygon whose vertices all lie on a circle is called an inscribed polygon. In this case, the circle is referred to as a circumscribed circle.
The pairs of opposite angles in an inscribed quadrilateral are supplementary.
In the diagram above, m∠A+m∠C=180∘andm∠B+m∠D=180∘. This can be proven using the Inscribed Angle Theorem.
Consider the circle and the inscribed quadrilateral ABCD.
Notice that the arcs BCD and BAD together span the entire circle. Together, the sum of their measures is 360∘. By the Inscribed Angle Theorem, the measure of arc BCD is equal to twice the measure of ∠A. Similarly, the measure of BAD is twice that of m∠C. By substitution, this gives 2m∠C+2m∠A=360∘. Simplifying gives m∠C+m∠A=180∘.
Similar reasoning yields the same relationship between m∠B and m∠D. Therefore, the pairs of opposite angles of an inscribed quadrilateral are supplementary.Using a compass and straightedge, it's possible to construct a square that's inscribed in a given circle. Start by drawing any diameter, naming the endpoints A and B.
Then, using the compass and straightedge, draw the perpendicular bisector of the diameter. Name the intersections between the bisector and the circle C and D, respectively.
The quadrilateral ACBD can now be drawn, which is in fact a square.
In a couple of steps, it's possible to draw the inscribed circle of a triangle, which is the largest circle completely contained within the triangle. First, draw the angle bisector of two vertices of the triangle. Label their intersection A.
This point, A, is called the incenter of the triangle and is the only point for which the shortest distance to each side is equal. Next, draw a line through A that intersects one of the sides at a right angle. Label this intersection B.
Place the sharp end of the compass at A and the pencil end at B. Draw a circle.
Finding the circumscribed circle of a triangle, which is the circle intersecting all vertices, is done in a similar fashion as finding the inscribed one. Start by drawing the perpendicular bisector of two sides of the triangle. Label their intersection A.
This point, A, is called the circumcenter and is located at equal distance from the vertices of the triangle. Now, place the sharp end of the compass in A, the pencil end in any vertex, and draw the corresponding circle.
